Question: Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{-4t^3 + 32t^2 - 48t}{7t^2 - 77t + 210}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ z = \dfrac {-4t(t^2 - 8t + 12)} {7(t^2 - 11t + 30)} $ $ z = -\dfrac{4t}{7} \cdot \dfrac{t^2 - 8t + 12}{t^2 - 11t + 30} $ Next factor the numerator and denominator. $ z = - \dfrac{4t}{7} \cdot \dfrac{(t - 6)(t - 2)}{(t - 6)(t - 5)}$ Assuming $t \neq 6$ , we can cancel the $t - 6$ $ z = - \dfrac{4t}{7} \cdot \dfrac{t - 2}{t - 5}$ Therefore: $ z = \dfrac{ -4t(t - 2)}{ 7(t - 5)}$, $t \neq 6$